A diverging lens of focal length 10 cm having refractive index 1.5 is immersed in a liquid of refractive index 3. The focal length and nature of the lens in liquid is

Options

A. )10 cm, convergent
B. 10 cm divergent
C. 18 cm, convergent
D. 72 cm, divergent

Explanation

To solve the problem of determining the focal length and nature of a diverging lens when it is immersed in a liquid, we can follow these steps:

### Step 1: Understand the given information
We have a diverging lens with:
- Focal length (f) = -10 cm (since it is a diverging lens)
- Refractive index of the lens (n_lens) = 1.5
- Refractive index of the liquid (n_liquid) = 3

### Step 2: Use the lens maker's formula
The lens maker's formula is given by:

\[ \frac{1}{f} = (n_{lens} - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( n_{lens} \) is the refractive index of the lens,
- \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

### Step 3: Calculate \( \frac{1}{R_1} - \frac{1}{R_2} \)
Since we know the focal length of the lens in air, we can rearrange the formula to find \( \frac{1}{R_1} - \frac{1}{R_2} \):

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Substituting the values:

\[ \frac{1}{-10} = (0.5) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

This simplifies to:

\[ \frac{1}{R_1} - \frac{1}{R_2} = -\frac{1}{5} \]

### Step 4: Determine the new focal length in the liquid
When the lens is immersed in a liquid, we can use the lens maker's formula again:

\[ \frac{1}{f'} = \left( \frac{n_{lens}}{n_{liquid}} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Substituting the values:

\[ \frac{1}{f'} = \left( \frac{1.5}{3} - 1 \right) \left( -\frac{1}{5} \right) \]

Calculating \( \frac{1.5}{3} - 1 \):

\[ \frac{1.5}{3} = 0.5 \implies 0.5 - 1 = -0.5 \]

Now substituting back:

\[ \frac{1}{f'} = (-0.5) \left( -\frac{1}{5} \right) = \frac{1}{10} \]

### Step 5: Calculate the new focal length
Taking the reciprocal gives:

\[ f' = 10 \, \text{cm} \]

### Step 6: Determine the nature of the lens
Since the focal length \( f' \) is positive, the lens behaves as a converging lens in the liquid.

### Final Answer
The focal length of the lens in the liquid is **10 cm**, and the nature of the lens is **converging**.

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